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%%文档的题目、作者与日期
%\author{王立庆（2021级数学与应用数学1班） }
\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
%\title{金融观点下的随机分析基础}
\title{第3章复习题 - 随机微分方程}
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%\date{2021 年 9 月 14 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}\itemsep1em

\item  设 $a,b$ 是两个连续函数，求解下述微分方程 $$dx = a(t)b(x)dt.$$

\vspace{4cm}

\item  解释下述积分方程等价于一个微分方程的初值问题，$$x(t)=x(0)+\int_0^t a(s,x(s))ds.$$

\vspace{4cm}

\item  设 $Y$ 是一个随机变量。解释下述随机微分方程的含义，$$dX_t = a(t,X_t)dt, \,\,\, X_0(\omega)=Y(\omega). $$

\vspace{5cm}
\newpage

\item  解释下述随机微分方程的含义，$$dX_t = a(t,X_t)dt + b(t,X_t)dB_t, \,\,\, X_0(\omega)=Y(\omega). $$

\vspace{7cm}

\item  使用 Ito 公式求解下述 Ito 随机微分方程，
$$X_t = X_0 + c\int_0^t X_s ds + \sigma \int_0^t X_sdB_s, \,\,\, t\in [0,T]. $$

\vspace{5cm}
\newpage

\item  求解下述 Ito 随机微分方程，%$$dX_t = cX_tdt + \sigma dB_t.$$
$$X_t = X_0 + c\int_0^t X_s ds + \sigma \int_0^t dB_s, \,\,\, t\in [0,T]. $$

\vspace{7cm}

\item  设 $B^{(1)}$ 和 $B^{(2)}$ 是两个相互独立的标准布朗运动，则下述定义的 $\tilde{B}$ 也是一个标准布朗运动，
$$\tilde{B}_t =  (\sigma_1^2+\sigma_2^2)^{-1/2} \left[\sigma_1B_t^{(1)} + \sigma_2 B_t^{(2)} \right].$$

\vspace{5cm}
\newpage

\item  求解下述随机微分方程，
$$X_t = X_0 + c\int_0^t X_sds + \sigma_1 \int_0^t X_sdB_s^{(1)} + \sigma_2 \int_0^t X_sdB_s^{(2)}. $$

\vspace{7cm}

\item  使用 Euler 方法和 Milstein 方法求下述随机微分方程的数值解，
$$X_t = X_0 + \int_0^t a(X_s) ds + \int_0^t b(X_s)dB_s, \,\,\, t\in [0,T]. $$

\vspace{5cm}

\end{enumerate}

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\end{document}

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